Darwin's Heights of Cross- and Self-fertilized Zea May Pairs

Description

Darwin (1876) studied the growth of pairs of zea may (aka corn)
seedlings, one produced by cross-fertilization and the other
produced by self-fertilization, but otherwise grown under identical
conditions.
His goal was to demonstrate the greater vigour of the cross-fertilized plants.
The data recorded are the final height (inches, to the nearest 1/8th) of the plants in each pair.

In the Design of Experiments, Fisher (1935) used these data to illustrate
a paired t-test (well, a one-sample test on the mean difference, cross - self).
Later in the book (section 21), he used this data to illustrate an early example of a non-parametric permutation
test, treating each paired difference as having (randomly) either a positive or negative sign.

Usage

data(ZeaMays)

Format

A data frame with 15 observations on the following 4 variables.

pair

pair number, a numeric vector

pot

pot, a factor with levels 1234

cross

height of cross fertilized plant, a numeric vector

self

height of self fertilized plant, a numeric vector

diff

cross - self for each pair

Details

In addition to the standard paired t-test,
several types of non-parametric tests can be contemplated:

(a) Permutation test, where the values of, say self are permuted and diff=cross - self
is calculated for each permutation. There are 15! permutations, but a reasonably
large number of random permutations would suffice. But this doesn't take the paired samples
into account.

(b) Permutation test based on assigning each abs(diff) a + or - sign, and calculating the mean(diff).
There are 2^{15} such possible values. This is essentially what Fisher
proposed. The p-value for the test is the proportion of absolute mean differences
under such randomization which exceed the observed mean difference.

(c) Wilcoxon signed rank test: tests the hypothesis that the median signed rank of the diff is zero,
or that the distribution of diff is symmetric about 0, vs. a location shifted alternative.

Source

Darwin, C. (1876). The Effect of Cross- and Self-fertilization in the Vegetable Kingdom,
2nd Ed. London: John Murray.

Andrews, D. and Herzberg, A. (1985) Data:
a collection of problems from many fields for the student and research worker.
New York: Springer. Data retrieved from: https://www.stat.cmu.edu/StatDat/